Another space where this is often used is the space of meromorphic functions.
This is similar to the holomorphic case, but instead of using the standard
metric for convergence we must use the spherical metric. That is if σσ\sigma
is the spherical metric, then want fn⁢(z)→f⁢(z)normal-→subscriptfnzfzf_{n}(z)\to f(z) uniformly on compact
subsets to mean that σ⁢(fn⁢(z),f⁢(z))σsubscriptfnzfz\sigma(f_{n}(z),f(z)) goes to 0 uniformly on compact
subsets.

Note that this is a classical definition that, while very often used, is not really consistent
with modern naming. In more modern language,
one would give a metric
on the space of continuous (holomorphic) functions that corresponds to
convergence on compact subsets and then you’d say “precompact set of functions” in such a metric space instead of saying “normal family of continuous (holomorphic) functions”. This added generality however
makes it more cumbersome to use since one would need to define the metric mentioned above.

References

1
John B. Conway.
Functions of One Complex Variable I.
Springer-Verlag, New York, New York, 1978.